Chaos-hyperchaos Transition Research Articles

Complex Dynamics of Rössler–Nikolov–Clodong O Hyperchaotic System: Analysis and Computations

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The analysis and numerical simulations show that chaos–hyperchaos transition in RNC-O system is coupled to change in the equilibria type as one large hyperchaotic attractor occurs. Moreover, we show that for this system, a zero-Hopf bifurcation is not possible. We also consider the cases when the divergence of the system is a constant and detected two families of exact solutions.

Chaos-hyperchaos transition in three identical quorum-sensing mean-field coupled ring oscillators.

We investigate the dynamics of three identical three-dimensional ring synthetic genetic oscillators (repressilators) located in different cells and indirectly globally coupled by quorum sensing whereby it is meant that a mechanism in which special signal molecules are produced that, after the fast diffusion mixing and partial dilution in the environment, activate the expression of a target gene, which is different from the gene responsible for their production. Even at low coupling strengths, quorum sensing stimulates the formation of a stable limit cycle, known in the literature as a rotating wave (all variables have identical waveforms shifted by one third of the period), which, at higher coupling strengths, converts to complex tori. Further torus evolution is traced up to its destruction to chaos and the appearance of hyperchaos. We hypothesize that hyperchaos is the result of merging the saddle-focus periodic orbit (or limit cycle) corresponding to the rotating wave regime with chaos and present considerations in favor of this conclusion.

Characterization of the chaos-hyperchaos transition based on return times.

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.

Chaos–hyperchaos transition in a class of models governed by Sommerfeld effect

The present paper points out the scenario of the hyperchaotic attractor formation (an attractor with at least two positive Lyapunov exponents) in a class of models governed by the Sommerfeld effects. The Sommerfeld effect is the result of the conservation law of energy. When a dynamical system is coupled to a power source, it acts like a energy sink for which a part of the source energy is spend to deform the system rather than increasing the drive speed. The Sommerfeld effect involves riddling bifurcation which explains the creation of the hyperchaotic attractor.

Hyperchaotic set in continuous chaos–hyperchaos transition

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.

The coupled dynamics of two particles with different limit sets

We consider a system of two coupled particles evolving in a periodic and spatially symmetric potential under the influence of external driving and damping. The particles are driven individually in such a way that in the uncoupled regime, one particle evolves on a chaotic attractor, while the other evolves on regular periodic attractors. Notably only the latter supports coherent particle transport. The influence of the coupling between the particles is explored, and in particular how it relates to the emergence of a directed current. We show that increasing the (weak) coupling strength subdues the current in a process, which in phase-space, is related to a merging crisis of attractors forming one large chaotic attractor in phase-space. Further, we demonstrate that complete current suppression coincides with a chaos-hyperchaos transition.

Revisiting the box counting algorithm for the correlation dimension analysis of hyperchaotic time series

We undertake the correlation dimension analysis of hyperchaotic time series using the box counting algorithm. We show that the conventional box counting scheme is inadequate for the accurate computation of correlation dimension ( D 2) of a hyperchaotic attractor and propose a modified scheme which is automated and gives better convergence of D 2 with respect to the number of data points. The scheme is first tested using the time series from standard chaotic systems, pure noise and data added with noise. It is then applied on the time series from three standard hyperchaotic systems for computing D 2. Our analysis clearly reveals that a second scaling region appears at lower values of box size as the system makes a transition into the hyperchaotic phase. This, in turn, suggests that correlation dimension analysis can also give information regarding chaos-hyperchaos transition.

Using recurrences to characterize the hyperchaos-chaos transition

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

Hyperchaos–chaos–Hyperchaos Transition in a Class of On–Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems' dynamics. After the bifurcation the phenomenon of on–off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos–chaos–hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.

Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems

Blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric subspace, becomes transversely unstable. There has been previous reports of chaos–hyperchaos transition via blowout bifurcation in synchronization of identical chaotic systems. In this paper, two five-dimensional continuous autonomous systems are considered, in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes of the systems’ dynamics. It is observed that one system undergoes a symmetric hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the dynamical behaviours before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole systems, which provides a possible route to hyperchaos.

Hyperchaos–chaos–hyperchaos transition in modified Rössler systems

Abstract We consider in this paper a family of modified hyperchaotic Rossler systems and investigate both problems of understanding hyperchaos–chaos–hyperchaos transition and computing the prediction time. These systems were obtained and numerically investigated by Nikolov and Clodong [Nikolov S, Clodong S. Occurrence of regular, chaotic and hyperchaotic behavior in a family of modified Rossler hyperchaotic systems. Chaos, Solitons & Fractals 2004;22:407–31]. Our studies confirm that transition hyperchaos–chaos–hyperchaos (i) depends on the change of the sign of the corresponding characteristic equation roots or (ii) can be obtained as a result of the absorption/repulsion of the repeller originally located out of the attractor by the growing attractor. It is also shown that the prediction time is a more reliable predictor of the evolution than the information dimension. We conclude that the prediction time in hyperchaotic regimes is at least one order of magnitude smaller than those in chaotic zones.

Manifestation of riddling in the presence of a small parameter mismatch between coupled systems.

Riddling bifurcation, i.e., the bifurcation in which one of the unstable periodic orbits embedded in a chaotic attractor becomes unstable transverse to the attractor, leads to the loss of chaos synchronization in coupled identical systems. We discuss here the manifestation of the riddling bifurcation for the case of a small parameter mismatch between coupled systems. We show that for slightly nonidentical coupled systems, the transverse growth of the synchronous attractor is mediated by transverse bifurcations of unstable periodic orbits embedded into the attractor. The desynchronization mechanism is shown to be similar to the case of chaos-hyperchaos transition.

Nonlinear dynamics and chaos in coupled shape memory oscillators

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. This contribution concerns with the dynamical response of coupled shape memory oscillators. Equations of motion are formulated by assuming a polynomial constitutive model to describe the restitution force of the oscillators and, since they are associated with a five-dimensional system, the analysis is performed by splitting the state space in subspaces. Free and forced vibrations are analyzed showing different kinds of responses. Periodic, quasi-periodic, chaos and hyperchaos are all possible in this system. Numerical investigations show interesting and complex behaviors. Dynamical jumps in free vibration and amplitude variation when temperature characteristics are changed are some examples. This article also shown some characteristics related to chaos–hyperchaos transition.

Symmetry-increasing bifurcation as a predictor of a chaos-hyperchaos transition in coupled systems.

In weakly coupled systems, it is possible to observe the coexistence of the chaotic attractors which are located out of the invariant manifold and are not symmetrical in relation to this manifold. When the control parameter is changed, these attractors can undergo a chaos-hyperchaos transition. We give numerical evidence that before this transition the coexisting attractors merge together creating an attractor symmetrical with respect to the invariant manifold. We argue that the attractors that are not located at the invariant manifold can exhibit dynamical behavior similar to bubbling and on-off intermittency previously observed for the attractors located at the invariant manifold, and we describe the mechanism of these phenomena.

Chaos–hyperchaos transition in coupled Rössler systems

Many of the nonlinear high-dimensional systems have hyperchaotic attractors. Typical trajectory on such attractors is characterized by at least two positive Lyapunov exponents. We provide numerical evidence that chaos–hyperchaos transition in six-dimensional dynamical system given by flow can be characterized by the set of infinite number of unstable periodic orbits embedded in the attractor as it was previously shown for the case of two coupled discrete maps.

Chaos-hyperchaos transition

The chaos-hyperchaos transition occurs when the second Lyapunov exponent becomes positive. We argue that this transition is mediated by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. Bifurcations of unstable periodic orbits occur in the neighborhood of the chaos-hyperchaos transition point where we observe unstable variable dimensionality. We give evidence that the chaos-hyperchaos transition is initiated by (i) the saddle-repeller bifurcation of a particular unstable periodic orbit usually of low period, (ii) the appearance of a repelling node in the saddle-node bifurcation, after which the chaotic attractor becomes riddled, or (iii) the absorption of the repeller (unstable node or focus) originally located out of the attractor by the growing attractor.

Chaos-hyperchaos transition

We discuss properties of an attractor in the neighbourhood of chaos-hyperchaos transition. The intermittency like model and a scaling law for the transition based on the features of the Poincaré map are developed. We investigate the properties of the Lyapunov and correlation dimensions in the neighbourhood of the transition point.

Transition to hyperchaos in the dynamics of a nonlinear vibration absorber

The problem of a qualitative distinction between chaotic and hyperchaotic responses in a vibration absorber with two degrees of freedom is considered here. Based on pertinent statistics of the Poincare' maps, some properties of the chaos-hyperchaos transition are examined and discussed.

HYPERCHAOTIC ATTRACTORS OF UNIDIRECTIONALLY-COUPLED CHUA’S CIRCUITS

In this letter we discuss properties of hyperchaotic attractors of unidirectionally-coupled Chua’s circuits. Results from chaos synchronization theory allowed us to observe chaos-hyperchaos transition on 3D projections of the attractor.

Transition to hyperchaos in chaotically forced coupled oscillators.

In this paper we discuss the Lyapunov and correlation dimensions of an attractor of a chain of unidirectionally coupled oscillators at the boundary of a chaos-hyperchaos transition. We discuss the problem of the distinction between chaos and hyperchaos based on these dimensions.